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Journal articles on the topic 'Nonlocal second order operators'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

Amster, P., and P. De Nápoli. "A nonlinear second order problem with a nonlocal boundary condition." Abstract and Applied Analysis 2006 (2006): 1–11. http://dx.doi.org/10.1155/aaa/2006/38532.

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We study a nonlinear problem of pendulum-type for ap-Laplacian with nonlinear periodic-type boundary conditions. Using an extension of Mawhin's continuation theorem for nonlinear operators, we prove the existence of a solution under a Landesman-Lazer type condition. Moreover, using the method of upper and lower solutions, we generalize a celebrated result by Castro for the classical pendulum equation.

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2

Euler, M., N. Euler, and M. C. Nucci. "On nonlocal symmetries generated by recursion operators: Second-order evolution equations." Discrete & Continuous Dynamical Systems - A 37, no. 8 (2017): 4239–47. http://dx.doi.org/10.3934/dcds.2017181.

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3

Baranets'kyy, Ya O. "Similitude operators generated by nonlocal problems for second-order elliptic equations." Ukrainian Mathematical Journal 44, no. 9 (September 1992): 1072–79. http://dx.doi.org/10.1007/bf01058366.

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4

Li, Meili, and Chunhai Kou. "Existence Results for Second-Order Impulsive Neutral Functional Differential Equations with Nonlocal Conditions." Discrete Dynamics in Nature and Society 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/641368.

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The existence of mild solutions for second-order impulsive semilinear neutral functional differential equations with nonlocal conditions in Banach spaces is investigated. The results are obtained by using fractional power of operators and Sadovskii's fixed point theorem.

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5

Kandemir, Mustafa. "SOLVABILITY OF BOUNDARY VALUE PROBLEMS WITH TRANSMISSION CONDITIONS FOR DISCONTINUOUS ELLIPTIC DIFFERENTIAL OPERATOR EQUATIONS." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 1 (March 30, 2016): 5842–57. http://dx.doi.org/10.24297/jam.v12i1.609.

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We consider nonlocal boundary value problems which includes discontinuous coefficients ellipticÂ differential operator equations of the second order and nonlocal boundary conditions together withÂ boundary-transmission conditions. We prove coerciveness and Fredholmness for these nonlocalÂ boundary value problems.

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6

MUSLIM, M., AVADHESH KUMAR, and RAVI P. AGARWAL. "Exact and trajectory controllability of second order nonlinear differential equations with deviated argument." Creative Mathematics and Informatics 26, no. 2 (2017): 181–91. http://dx.doi.org/10.37193/cmi.2017.02.07.

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In this manuscript, we consider a control system governed by a second order nonlinear differential equations with deviated argument in a Hilbert space X. We used the strongly continuous cosine family of bounded linear operators and ﬁxed point method to study the exact and trajectory controllability. Also, we study the exact controllability of the nonlocal control problem. Finally, we give an example to illustrate the application of these results.

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7

SHAW, JIIN-CHANG, and MING-HSIEN TU. "NONLOCAL EXTENDED CONFORMAL ALGEBRAS ASSOCIATED WITH MULTICONSTRAINT KP HIERARCHY AND THEIR FREE FIELD REALIZATIONS." International Journal of Modern Physics A 13, no. 16 (June 30, 1998): 2723–37. http://dx.doi.org/10.1142/s0217751x98001384.

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We study the conformal properties of the multiconstraint KP hierarchy and its non-standard partner by covariantizing their corresponding Lax operators. The associated second Hamiltonian structures turn out to be nonlocal extensions of Wn algebra by some integer or half-integer spin fields depending on the order of the Lax operators. In particular, we show that the complicated second Hamiltonian structure of the nonstandard multiconstraint KP hierarchy can be simplified by factorizing its Lax operator to multiplication form. We then diagonalize this simplified Poisson matrix and obtain the free field realizations of its associated nonlocal algebras.

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8

Anthoni, S. Marshal, J. H. Kim, and J. P. Dauer. "Existence of mild solutions of second-order neutral functional differential inclusions with nonlocal conditions in Banach spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 22 (2004): 1133–49. http://dx.doi.org/10.1155/s0161171204310410.

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We study the existence of mild solutions of the nonlinear second-order neutral functional differential and integrodifferential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using the theory of strongly continuous cosine families of bounded linear operators and a fixed point theorem for condensing maps due to Martelli.

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9

Mokin, A. Yu. "Spectral properties of a nonlocal second-order difference operator." Differential Equations 50, no. 7 (July 2014): 938–46. http://dx.doi.org/10.1134/s001226611407009x.

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10

Todorov, Todor D. "Nonlocal problem for a general second-order elliptic operator." Computers & Mathematics with Applications 69, no. 5 (March 2015): 411–22. http://dx.doi.org/10.1016/j.camwa.2014.12.014.

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11

Shakhmurov, Veli B. "Maximal regular boundary value problems in Banach-valued function spaces and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–26. http://dx.doi.org/10.1155/ijmms/2006/92134.

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The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

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12

Sajavičius, S. "On the eigenvalue problems for differential operators with coupled boundary conditions." Nonlinear Analysis: Modelling and Control 15, no. 4 (October 25, 2010): 493–500. http://dx.doi.org/10.15388/na.15.4.14320.

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In the paper, the eigenvalue problems for one- and two-dimensional second order differential operators with nonlocal coupled boundary conditions are considered. Conditions for the existence of zero, positive, negative or complex eigenvalues are proposed and analytical expressions of eigenvalues are provided.

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13

Freitas, Pedro, and Guido Sweers. "Positivity results for a nonlocal elliptic equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 4 (1998): 697–715. http://dx.doi.org/10.1017/s0308210500021727.

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In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

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14

Diem, Dang Huan. "Existence for a Second-Order Impulsive Neutral Stochastic Integrodifferential Equations with Nonlocal Conditions and Infinite Delay." Chinese Journal of Mathematics 2014 (February 27, 2014): 1–13. http://dx.doi.org/10.1155/2014/143860.

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The current paper is concerned with the existence of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in a Hilbert space. A sufficient condition for the existence results is obtained by using the Krasnoselskii-Schaefer-type fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. Finally, an application to the stochastic nonlinear wave equation with infinite delay is given.

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15

Tersenov, Ar S. "Solvability of the lyapunov equation for nonselfadjoint second-order differential operators with nonlocal boundary conditions." Siberian Mathematical Journal 39, no. 5 (October 1998): 1026–42. http://dx.doi.org/10.1007/bf02672926.

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16

Bonito, Andrea, Wenyu Lei, and Abner J. Salgado. "Finite element approximation of an obstacle problem for a class of integro–differential operators." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 1 (January 2020): 229–53. http://dx.doi.org/10.1051/m2an/2019058.

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We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

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17

Gorodetskyi, Vasyl V., Olga V. Martynyuk, and Olesia V. Feduh. "The well-posedness of a nonlocal multipoint problem for a differential operator equation of second order." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 67–79. http://dx.doi.org/10.1515/gmj-2018-0007.

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AbstractWe establish the well-posedness of a nonlocal multipoint problem for a second-order evolution equation with respect to a time variable with an operator having a discrete spectrum. A nonlocal condition is considered to be satisfied in a weak sense in the space of formal Fourier series that are identified with continuous linear functionals (generalized elements) on some space connected with the operator.

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18

Ashyralyev, Allaberen, and Ozgur Yildirim. "A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem." Abstract and Applied Analysis 2012 (2012): 1–29. http://dx.doi.org/10.1155/2012/846582.

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The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceHwith the self-adjoint positive definite operatorA. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

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19

Polyakov, D. M. "Nonlocal Perturbation of a Periodic Problem for a Second-Order Differential Operator." Differential Equations 57, no. 1 (January 2021): 11–18. http://dx.doi.org/10.1134/s001226612101002x.

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20

Čiegis, Raimondas, Remigijus Čiegis, and Ignas Dapšys. "A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators." Mathematics 9, no. 12 (June 10, 2021): 1344. http://dx.doi.org/10.3390/math9121344.

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The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

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21

Dovletov, Dovlet M. "Nonlocal boundary value problem in terms of flow for Sturm-Liouville operator in differential and difference statements." e-Journal of Analysis and Applied Mathematics 1, no. 1 (January 1, 2018): 37–55. http://dx.doi.org/10.2478/ejaam-2018-0004.

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AbstractSturm-Liouville operator with second kind of nonlocal boundary value conditions is considered. For the classical solution, a priori estimate is established and unique existence is proved. Associated finite-difference scheme is proposed on uniform mesh and second-order accuracy for approximation is proved. An application of obtained results to nonlocal boundary problems with weight integral conditions is provided.

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22

Khatimtsov, N. A., and F. E. Lomovtsev. "Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains." Differential Equations 47, no. 4 (April 2011): 503–15. http://dx.doi.org/10.1134/s0012266111040069.

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23

Kerimov, K. A., and S. S. Mirzoev. "On a problem for operator-differential second-order equations with nonlocal boundary condition." Mathematical Notes 94, no. 3-4 (September 2013): 330–34. http://dx.doi.org/10.1134/s0001434613090046.

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24

Ben-Ari, Iddo, and Ross G. Pinsky. "Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure." Journal of Functional Analysis 251, no. 1 (October 2007): 122–40. http://dx.doi.org/10.1016/j.jfa.2007.05.019.

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25

Budochkina, Svetlana A., and Ekaterina S. Dekhanova. "ON THE POTENTIALITY OF A CLASS OF OPERATORS RELATIVE TO LOCAL BILINEAR FORMS." Ural Mathematical Journal 7, no. 1 (July 30, 2021): 26. http://dx.doi.org/10.15826/umj.2021.1.003.

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The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

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26

Munoz, Ana Isabel, and Jose Ignacio Tello. "MATHEMATICAL ANALYSIS AND NUMERICAL SIMULATION IN MAGNETIC RECORDING." Mathematical Modelling and Analysis 19, no. 3 (June 1, 2014): 334–46. http://dx.doi.org/10.3846/13926292.2014.924081.

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The head-tape interaction in magnetic recording is described in the literature by a coupled system of partial differential equations. In this paper we study the limit case of the system which reduces the problem to a second order nonlocal equation on a one-dimensional domain. We describe the numerical method of resolution of the problem, which is reformulated as an obstacle one to prevent head-tape contact. A finite element method and a duality algorithm handling Yosida approximation tools for maximal monotone operators are used in order to solve numerically the obstacle problem. Numerical simulations are introduced to describe some qualitative properties of the solution. Finally some conclusions are drawn.

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27

Benchohra, M., and S. K. Ntouyas. "Controllability of Second-Order Differential Inclusions in Banach Spaces with Nonlocal Conditions." Journal of Optimization Theory and Applications 107, no. 3 (December 2000): 559–71. http://dx.doi.org/10.1023/a:1026447232030.

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28

Koksal, Mehmet Emir. "Recent Developments on Operator-Difference Schemes for Solving Nonlocal BVPs for the Wave Equation." Discrete Dynamics in Nature and Society 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/210261.

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The second-order one-dimensional linear hyperbolic equation with time and space variable coefficients and nonlocal boundary conditions is solved by using stable operator-difference schemes. Two new second-order difference schemes recently appeared in the literature are compared numerically with each other and with the rather old first-order difference scheme all to solve abstract Cauchy problem for hyperbolic partial differential equations with time-dependent unbounded operator coefficient. These schemes are shown to be absolutely stable, and the numerical results are presented to compare the accuracy and the execution times. It is naturally seen that the second-order difference schemes are much more advantages than the first-order ones. Although one of the second-order difference scheme is less preferable than the other one according to CPU (central processing unit) time consideration, it has superiority when the accuracy weighs more importance.

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29

BUCHBINDER, I. L., E. S. FRADKIN, S. L. LYAKHOVICH, and V. D. PERSHIN. "GENERALIZED CANONICAL QUANTIZATION OF BOSONIC STRING IN BACKGROUND FIELDS." International Journal of Modern Physics A 06, no. 07 (March 20, 1991): 1211–31. http://dx.doi.org/10.1142/s0217751x91000630.

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The theory of a closed bosonic string interacting with background fields, namely, with metric, antisymmetric tensor of second rank and dilaton, is considered. The classical formulation of the constrained theory is developed and the canonical quantization is carried out. The combined symbols of the Virasoro operators corresponding to the Weyl ordering of zero string modes and to the Wick ordering of oscillating ones are constructed. The quantum Virasoro algebra is formulated by means of star-commutators corresponding to these symbols and the nonlocal first quantum correction in the algebra is calculated. The local part of the correction linear in curvature is derived and effective equations of motion for background fields are obtained in the lowest order.

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30

Yuldashev, Tursun K., and Bakhtiyor J. Kadirkulovich. "NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR." Ural Mathematical Journal 6, no. 1 (July 29, 2020): 153. http://dx.doi.org/10.15826/umj.2020.1.013.

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In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

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31

Dovletov, D. M. "Nonlocal boundary value problem with Poissons operator on a rectangle and its difference interpretation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 38–54. http://dx.doi.org/10.31489/2020m3/38-54.

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In the present paper, differential and difference variants of nonlocal boundary value problem (NLBVP) for Poisson’s equation in open rectangular domain are studied. The existence, uniqueness and a priori estimate of classical solution are established. The second order of accuracy difference scheme is presented. The applications with weighted integral condition are provided in differential and difference variants.

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32

Andrianov, I. V., J. Awrejcewicz, and A. O. Ivankov. "On an elastic dissipation model as continuous approximation for discrete media." Mathematical Problems in Engineering 2006 (2006): 1–8. http://dx.doi.org/10.1155/mpe/2006/27373.

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Construction of an accurate continuous model for discrete media is an important topic in various fields of science. We deal with a 1D differential-difference equation governing the behavior of ann-mass oscillator with linear relaxation. It is known that a string-type approximation is justified for low part of frequency spectra of a continuous model, but for free and forced vibrations a solution of discrete and continuous models can be quite different. A difference operator makes analysis difficult due to its nonlocal form. Approximate equations can be obtained by replacing the difference operators via a local derivative operator. Although application of a model with derivative of more than second order improves the continuous model, a higher order of approximated differential equation seriously complicates a solution of continuous problem. It is known that accuracy of the approximation can dramatically increase using Padé approximations. In this paper, one- and two-point Padé approximations suitable for justify choice of structural damping models are used.

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33

Benchohra, M., and S. K. Ntouyas. "Controllability for an Infinite-Time Horizon of Second-Order Differential Inclusions in Banach Spaces with Nonlocal Conditions." Journal of Optimization Theory and Applications 109, no. 1 (April 2001): 85–98. http://dx.doi.org/10.1023/a:1017561821201.

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34

Bazazzadeh, Soheil, Arman Shojaei, Mirco Zaccariotto, and Ugo Galvanetto. "Application of the peridynamic differential operator to the solution of sloshing problems in tanks." Engineering Computations 36, no. 1 (December 21, 2018): 45–83. http://dx.doi.org/10.1108/ec-12-2017-0520.

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PurposeThe purpose of this paper is to apply the Peridynamic differential operator (PDDO) to incompressible inviscid fluid flow with moving boundaries. Based on the potential flow theory, a Lagrangian formulation is used to cope with non-linear free-surface waves of sloshing water in 2D and 3D rectangular and square tanks.Design/methodology/approachIn fact, PDDO recasts the local differentiation operator through a nonlocal integration scheme. This makes the method capable of determining the derivatives of a field variable, more precisely than direct differentiation, when jump discontinuities or gradient singularities come into the picture. The issue of gradient singularity can be found in tanks containing vertical/horizontal baffles.FindingsThe application of PDDO helps to obtain the velocity field with a high accuracy at each time step that leads to a suitable geometry updating for the procedure. Domain/boundary nodes are updated by using a second-order finite difference time algorithm. The method is applied to the solution of different examples including tanks with baffles. The accuracy of the method is scrutinized by comparing the numerical results with analytical, numerical and experimental results available in the literature.Originality/valueBased on the investigations, PDDO can be considered a reliable and suitable approach to cope with sloshing problems in tanks. The paper paves the way to apply the method for a wider range of problems such as compressible fluid flow.

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35

Correa, Ernesto, and Arturo de Pablo. "Nonlocal operators of order near zero." Journal of Mathematical Analysis and Applications 461, no. 1 (May 2018): 837–67. http://dx.doi.org/10.1016/j.jmaa.2017.12.011.

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36

Tian, Xiaochuan, and Qiang Du. "A Class of High Order Nonlocal Operators." Archive for Rational Mechanics and Analysis 222, no. 3 (July 6, 2016): 1521–53. http://dx.doi.org/10.1007/s00205-016-1025-8.

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37

Bell, Denis R., and Salah-Eldin A. Mohammed. "degenerate second-order operators." Duke Mathematical Journal 78, no. 3 (June 1995): 453–75. http://dx.doi.org/10.1215/s0012-7094-95-07822-3.

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38

Andersson, Lars, Thomas Bäckdahl, and Pieter Blue. "Second order symmetry operators." Classical and Quantum Gravity 31, no. 13 (June 17, 2014): 135015. http://dx.doi.org/10.1088/0264-9381/31/13/135015.

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39

Casati, M., E. V. Ferapontov, M. V. Pavlov, and R. F. Vitolo. "On a class of third-order nonlocal Hamiltonian operators." Journal of Geometry and Physics 138 (April 2019): 285–96. http://dx.doi.org/10.1016/j.geomphys.2018.10.018.

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40

Löbus, J. U. "Generalized Second Order Differential Operators." Mathematische Nachrichten 152, no. 1 (1991): 229–45. http://dx.doi.org/10.1002/mana.19911520119.

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41

Byszewski, Ludwik, and Teresa Winiarska. "An abstract nonlocal second order evolution problem." Opuscula Mathematica 32, no. 1 (2012): 75. http://dx.doi.org/10.7494/opmath.2012.32.1.75.

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42

Benchohra, Mouffak, Juan J. Nieto, and Noreddine Rezoug. "Second order evolution equations with nonlocal conditions." Demonstratio Mathematica 50, no. 1 (December 20, 2017): 309–19. http://dx.doi.org/10.1515/dema-2017-0029.

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Abstract In this paper, we shall establish sufficient conditions for the existence of solutions for second order semilinear functional evolutions equation with nonlocal conditions in Fréchet spaces. Our approach is based on the concepts of Hausdorff measure, noncompactness and Tikhonoff’s fixed point theorem. We give an example for illustration.

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43

Benedetti, I., N. V. Loi, L. Malaguti, and V. Taddei. "Nonlocal diffusion second order partial differential equations." Journal of Differential Equations 262, no. 3 (February 2017): 1499–523. http://dx.doi.org/10.1016/j.jde.2016.10.019.

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44

Blower, Gordon. "Hankel operators that commute with second-order differential operators." Journal of Mathematical Analysis and Applications 342, no. 1 (June 2008): 601–14. http://dx.doi.org/10.1016/j.jmaa.2007.12.028.

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45

Kenig, C. E., A. Ruiz, and C. D. Sogge. "second order constant coefficient differential operators." Duke Mathematical Journal 55, no. 2 (June 1987): 329–47. http://dx.doi.org/10.1215/s0012-7094-87-05518-9.

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46

ter Elst, A. F. M., Derek W. Robinson, Adam Sikora, and Yueping Zhu. "Second-order operators with degenerate coefficients." Proceedings of the London Mathematical Society 95, no. 2 (April 24, 2007): 299–328. http://dx.doi.org/10.1112/plms/pdl017.

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47

Behncke, Horst, and Don B. Hinton. "ᵉe-symmetric second order differential operators." Operators and Matrices, no. 4 (2020): 871–908. http://dx.doi.org/10.7153/oam-2020-14-54.

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48

Todorov, Todor. "A second order problem with nonlocal boundary conditions." Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics 10, no. 2 (2018): 71–74. http://dx.doi.org/10.5937/spsunp1802071t.

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49

Mawhin, Jean, Bogdan Przeradzki, and Katarzyna Szymańska-Dȩbowska. "Second order systems with nonlinear nonlocal boundary conditions." Electronic Journal of Qualitative Theory of Differential Equations, no. 56 (2018): 1–11. http://dx.doi.org/10.14232/ejqtde.2018.1.56.

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50

Franco, Daniel, Gennaro Infante, and Mirosława Zima. "Second order nonlocal boundary value problems at resonance." Mathematische Nachrichten 284, no. 7 (April 14, 2011): 875–84. http://dx.doi.org/10.1002/mana.200810841.

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